MICROSCOPE. 



333 



within the principal focus, they will neither be 

 brought to converge nor be rendered parallel, 

 but will diverge in a diminished degree. The 

 same principles apply equally to a plano- 

 convex lens, the distance of its principal focus 

 being understood to be the diameter of the 

 sphere. They also apply to a lens whose sur- 

 faces have different curvatures ; the principal 

 focus of such a lens is found by multiplying the 

 radius of one surface by the radius of the other, 

 and dividing this product by half the sum of 

 the same radii. For the rules by which the 

 foci of convex lenses may be found for rays of 

 different degrees of convergence and divergence, 

 we must refer to works on optics. 



The influence of concave lenses will evidently 



Fig. 150. 



Fig. 153. 



Parallel rays falling on a plano-concave lens made to 

 diverge as from its principal focus, and rays con- 

 verging to that focus rendered parallel. 



be precisely the converse of that of convex. 

 Rays which fall upon them in a parallel direc- 

 tion will be made to diverge as if from the 

 principal focus, which is here called the nega- 

 tive focus. This will be, for a plano-concave 



Fig. 151. 



Parallel rays made to diverge as from the principal 

 focus, and rays converging to that focus rendered 

 parallel. 



Fig. 152. 



Rays slightly converging made to diverge. 



if uninterrupted, they would have met in the 

 principal focus, will be rendered parallel ; if 

 converging more, they will still meet, but at a 

 greater distance ; and if converging less, they 

 will diverge as from a negative focus at a greater 

 distance than that for parallel rays. If already 

 diverging, they will diverge still more, as from 

 a negative focus nearer than the principal focus; 

 but this will approach the principal focus, in 

 proportion as the distance of the point of di- 

 vergence is such, that the direction of the rays 

 approaches the parallel. 



If a lens be convex on one side and concave 

 on the other, forming what is called a meniscus, 

 its effect will depend upon the proportion be- 

 tween the two curvatures. If they are equal, as 

 in a watch-glass, no perceptible effect will be 

 produced ; if the convex curvature be the 

 greater, the effect will be that of a less powerful 

 convex lens ; and if the concave curvature be 

 the more considerable, it will be that of a less 

 powerful concave lens. The focus of conver- 

 gence for parallel rays in the first case, and of 

 divergence in the second, may be found by 

 dividing the product of the two radii by half 

 their difference. 



Hitherto we have considered only the effects 

 of lenses upon a pencil of rays issuing from a 

 single luminous point, and that point situated 

 in the line of its axis. If the point be situated 

 above the line of its axis, the focus will be 

 below it, and vice versa. The surface of every 

 luminous body may be regarded as comprehend- 

 ing an infinite number of such points, from 

 every one of which a pencil of rays proceeds, 

 and is refracted according to the laws already 

 specified ; so that a perfect but inverted image 

 or picture of the object is formed upon any 

 surface placed in the focus, and adapted to re- 

 ceive the rays. 



Fig. 154. 



Hays greatly converging made to converge less, and 

 rays slightly diverging made to diverge more. 



lens, at the distance of the diameter of the sphere 

 of curvature; and for a double concave, in the 

 centre of that sphere. In the same manner, 

 rays which are converging to such a degree that, 



In optical diagrams it is usual, in order to 

 avoid confusion, to mark out the course of the 

 rays proceeding from two or three only of such 

 points. By an inspection of the subjoined 

 figures, it will be evident that, if the object be 

 placed at twice the distance of the principal 

 focus, the image being formed at an equal dis- 



